Fractal Percolation, Porosity, and Dimension

Abstract

We study the porosity properties of fractal percolation sets $$E\subset \mathbb {R}^d$$ . Among other things, for all $$0<\varepsilon <\tfrac{1}{2}$$ , we obtain dimension bounds for the set of exceptional points where the upper porosity of E is less than $$\tfrac{1}{2}-\varepsilon $$ , or the lower porosity is larger than $$\varepsilon $$ . Our method works also for inhomogeneous fractal percolation and more general random sets whose offspring distribution gives rise to a Galton–Watson process.